Calculate the angle of repose from a pile’s height and base radius, or find pile height or radius from the angle, with slope, volume and powder flow class.
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Angle of Repose Formula
- θ = angle of repose, in degrees
- h = vertical height of the pile
- r = horizontal run from the center of the pile to the edge of the base (half the base width for a symmetric cone)
The ideal friction mode uses μ = tan(θ) for a simple body on an inclined plane at incipient slip. It does not measure particle or internal friction from a pile angle. Select pile height or base radius to solve h = r · tan(θ) or r = h / tan(θ).
Assumptions: the pile is a symmetric cone or wedge formed by free-pouring loose material onto a flat surface, particle properties are uniform, and no external compaction or vibration is applied. Moisture, particle shape, and packing can shift real-world results.
Angle and Ideal-Coefficient Examples
These mathematical examples show tan(θ). They are not measured material properties or design limits.
| Geometry example | Angle | μ ≈ tan(θ) |
|---|---|---|
| Ideal 20° case | 20° | 0.36397 |
| Ideal 30° case | 30° | 0.57735 |
| Ideal 35° case | 35° | 0.70021 |
| Ideal 40° case | 40° | 0.83910 |
| Ideal 45° case | 45° | 1.00000 |
The next table describes slope geometry only. Angle alone cannot determine cohesion or moisture; the powder flow class shown for pharmaceutical powders follows the USP <1174> angle ranges.
| Angle | Behavior |
|---|---|
| < 20° | Shallow geometric slope. |
| 20°–30° | Low-to-moderate geometric slope. |
| 30°–40° | Moderate geometric slope. |
| 40°–50° | Steep geometric slope. |
| > 50° | Very steep geometric slope; material behavior requires separate testing. |
Example
You measure a sand pile that is 2 ft tall with a 7 ft base diameter. The horizontal run is 7 / 2 = 3.5 ft.
- θ = arctan(2 / 3.5) = arctan(0.571) ≈ 29.7°
- μ = tan(29.7°) ≈ 0.57
This is a geometric estimate. It cannot identify moisture, material condition, or slope stability from the angle alone.
FAQ
Do I use the diameter or the radius? The formula needs the horizontal run from the peak to the edge. If you measured the full base diameter, divide it by two before entering the base radius.
Why is my measured angle different from the table? Moisture, particle shape, fines content, drop height, and surface vibration all change the result. Tables list typical ranges, not fixed values.
Is angle of repose the same as the friction angle? They are closely related but not identical. Angle of repose is measured on a free-formed pile; the internal friction angle is measured in a controlled shear test. No universal offset between the two angles is assumed here; material and test method matter.
